Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size

Abstract

We construct affinization of the algebra $g{l}_{\lambda }$ of ``complex size'' matrices, that contains the algebras $\widehat{g{l}_n}$ for integral values of the parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra $\widehat{g{l}_{\lambda }}$ results in the quadratic Gelfand--Dickey structure on the Poisson--Lie group of all pseudodifferential operators of fractional order. This construction is extended to the simultaneous deformation of orthogonal and simplectic algebras that produces self-adjoint operators, and it has a counterpart for the Toda lattices with fractional number of particles.

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